Question

The determinant $\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|$ is divisible by.

• A. $x-y$
• B. $x^2-y^2+xy$
• C. $x^2+xy+y^2$
• D. $x^2-xy+y^2$

Answer 1

The correct option is D $x^2-xy+y^2$

We have,

$\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|$

using row transformation,

$R_3\to R_3-(R_1+R_2)$

$\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ 0& -2y& -2x\end{array}\right|$

$=2y[x^2-(x+y\left)y\right]-2x\left[x\right(x+y)-y^2]$

$=2y{x}^2-2y^{2}x-2y^3-2x^3-2x^{2}y+2{xy}^2$

Post cancelling,

$=-2(y^3+x^3)$

$=-2(x+y)(x^2+y^2-xy)$

as $x^2+y^2-xy$ is a factor of its determinants

$\therefore$ $x^2+y^2-xy$ will divide the above determinant.